Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

Given a totally real number field $F$ and its adèle ring $\mathbb{A}_{F}$, let $\pi$ vary in the set of irreducible cuspidal automorphic representations of $\operatorname{PGL}_{2}(\mathbb{A}_{F})$ corresponding to primitive Hilbert modular forms of a fixed weight. We determine the symmetry type of the one-level density of low-lying zeros of the symmetric power $L$-functions $L(s, \operatorname{Sym}^{r}(\pi))$ weighted by special values of the symmetric square $L$-functions $L((z+1)/2, \operatorname{Sym}^{2}(\pi))$ at $z \in [0, 1]$ in the level aspect. If $0 < z \leq 1$, our weighted density in the level aspect has the same symmetry type as Ricotta and Royer's density of low-lying zeros of symmetric power $L$-functions for $F = \mathbb{Q}$ with harmonic weight. Hence our result is regarded as a $z$-interpolation of Ricotta and Royer's result. If $z = 0$, the density of low-lying zeros weighted by central values is of a different type only when $r = 2$.